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<DL>
<DT><A NAME="foot3925">... defined</A><A NAME="foot3925"
 HREF="node36.html#tex2html383"><SUP>2.1</SUP></A>
<DD>If we tried
to compute the trivial eigenvalues in the same way as the nontrivial
ones, that is by taking ratios of the leading <I>n</I>-<I>r</I> diagonal entries
of <I>X</I><SUP><I>T</I></SUP> <I>A</I><SUP><I>T</I></SUP> <I>AX</I> and <I>X</I><SUP><I>T</I></SUP> <I>B</I><SUP><I>T</I></SUP> <I>B X</I>, we would get 0/0. For a detailed
mathematical discussion of this decomposition, see the discussion of
the Kronecker Canonical Form in [<A
 HREF="node151.html#gantmacher">53</A>].
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</PRE>
<DT><A NAME="foot3940">...
xTGSJA</A><A NAME="foot3940"
 HREF="node59.html#tex2html1653"><SUP>2.2</SUP></A>
<DD>
If <B><I>m</I>-<I>k</I>-<I>l</I> &lt; 0</B>, we may add some zero rows to <B><I>A</I><SUB>23</SUB></B> to make
it upper triangular.
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</PRE>
<DT><A NAME="foot9747">... digit</A><A NAME="foot9747"
 HREF="node74.html#tex2html1856"><SUP>4.1</SUP></A>
<DD>This is the
case on Cybers, Cray X-MP, Cray Y-MP, Cray&nbsp;2 and Cray C90.
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</PRE>
<DT><A NAME="foot13097">... xLAMCH</A><A NAME="foot13097"
 HREF="node74.html#tex2html1857"><SUP>4.2</SUP></A>
<DD>See subsection&nbsp;<A HREF="node24.html#subsecnaming">2.2.3</A> for explanation
of the naming convention used for LAPACK routines.
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</PRE>
<DT><A NAME="foot9755">...
PCs</A><A NAME="foot9755"
 HREF="node74.html#tex2html1858"><SUP>4.3</SUP></A>
<DD>Important machines that do not implement the IEEE standard
include the Cray XMP, Cray YMP, Cray 2, Cray C90, IBM 370 and DEC Vax.
Some architectures have two (or more) modes, one that implements IEEE
arithmetic, and another that is less accurate but faster.
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</PRE>
<DT><A NAME="foot9804">... xSTEVR.</A><A NAME="foot9804"
 HREF="node74.html#tex2html1860"><SUP>4.4</SUP></A>
<DD>xSYEVR and the
         other drivers call ILAENV to
         check if IEEE arithmetic is available, and uses another algorithm if it is not.
         If LAPACK is installed by following the directions in Chapter&nbsp;6, then
         ILAENV will return the correct information about the availability of
         IEEE arithmetic. If xSYEVR or the other drivers are used without this
         installation procedure,
         then the default is for ILAENV to say the IEEE arithmetic is available,
         since this is most common and faster.
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</PRE>
<DT><A NAME="foot10322">....</A><A NAME="foot10322"
 HREF="node78.html#tex2html2014"><SUP>4.5</SUP></A>
<DD>
Sometimes our algorithms satisfy only 
<!-- MATH
 ${\rm alg}(z)=f(z+ \delta) + \eta$
 -->
<IMG
 WIDTH="166" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img369.gif"
 ALT="${\rm alg}(z)=f(z+ \delta) + \eta$">
where both
<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img366.gif"
 ALT="$\delta$">
and <IMG
 WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img370.gif"
 ALT="$\eta$">
are small. This does not significantly change the following
analysis.
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</PRE>
<DT><A NAME="foot10323">... small</A><A NAME="foot10323"
 HREF="node78.html#tex2html2015"><SUP>4.6</SUP></A>
<DD>More generally,
we only need Lipschitz continuity of <B><I>f</I></B>, and may use the Lipschitz
constant in place of <B><I>f</I>'</B> in deriving error bounds.
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</PRE>
<DT><A NAME="foot10349">... exist)</A><A NAME="foot10349"
 HREF="node78.html#tex2html2016"><SUP>4.7</SUP></A>
<DD>This is
a different use of the term ill-posed than used in other contexts. For
example, to be well-posed (not ill-posed) in the sense of Hadamard,
it is sufficient for <B><I>f</I></B> to be continuous,
whereas we require Lipschitz continuity.
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</PRE>
<DT><A NAME="foot13140">... described</A><A NAME="foot13140"
 HREF="node78.html#tex2html2017"><SUP>4.8</SUP></A>
<DD>There are some
caveats to this statement. When computing the inverse of a matrix,
the backward error <B><I>E</I></B> is small taking the
columns of the computed inverse one at
a time, with a different <B><I>E</I></B> for each column [<A
 HREF="node151.html#lapwn27">47</A>].
The same is true when computing
the eigenvectors of a nonsymmetric matrix.
When computing the eigenvalues and eigenvectors
of <IMG
 WIDTH="63" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img56.gif"
 ALT="$A - \lambda B$">,
<IMG
 WIDTH="72" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img383.gif"
 ALT="$AB- \lambda I$">
or <IMG
 WIDTH="72" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img384.gif"
 ALT="$BA- \lambda I$">,
with <B><I>A</I></B> symmetric and <B><I>B</I></B> symmetric and positive definite
(using xSYGV or xHEGV) then the method may not be backward normwise stable if
<A NAME="10357"></A><A NAME="10358"></A><A NAME="10359"></A><A NAME="10360"></A>
<B><I>B</I></B> has a large condition number 
<!-- MATH
 $\kappa_{\infty}(B)$
 -->
<IMG
 WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img385.gif"
 ALT="$\kappa_{\infty}(B)$">,
although it has useful error bounds in this case too
(see section <A HREF="node98.html#secgendef">4.10</A>). Solving the Sylvester equation<A NAME="10362"></A>
<B><I>AX</I>+<I>XB</I>=<I>C</I></B> for the matrix <B><I>X</I></B> may not be backward stable, although
there are again useful error bounds for <B><I>X</I></B> [<A
 HREF="node151.html#higham93">66</A>].
<PRE>.
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</PRE>
<DT><A NAME="foot13141">...
error</A><A NAME="foot13141"
 HREF="node79.html#tex2html2039"><SUP>4.9</SUP></A>
<DD>For other algorithms, the answers (and computed error bounds)
are as accurate as though the algorithms were componentwise relatively backward
stable, even though they are not. These algorithms are called
<EM> componentwise relatively forward stable</EM>.
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</PRE>
<DT><A NAME="foot13155">... bound</A><A NAME="foot13155"
 HREF="node80.html#tex2html2046"><SUP>4.10</SUP></A>
<DD>As discussed in
section&nbsp;<A HREF="node75.html#secnormnotation">4.2</A>, this approximate error bound
may underestimate the true error by a factor <B><I>p</I>(<I>n</I>)</B>
which is a modestly growing function of the problem dimension <B><I>n</I></B>.
Often <IMG
 WIDTH="88" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img395.gif"
 ALT="$p(n) \leq 10n$">.
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</PRE>
<DT><A NAME="foot13159">... suppose</A><A NAME="foot13159"
 HREF="node80.html#tex2html2047"><SUP>4.11</SUP></A>
<DD>This and other numerical examples
were computed in IEEE single precision arithmetic [<A
 HREF="node151.html#ieee754">4</A>]
on a DEC 5000/120 workstation.
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</PRE>
<DT><A NAME="fnm\valuefootnoteapproxerrorbound">...
bound<SUP>4.10</SUP></A>
<DD>
<PRE>.
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</PRE>
<DT><A NAME="fnm\valuefootnoteDEC5000">... example<SUP>4.11</SUP></A>
<DD>
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</PRE>
<DT><A NAME="fnm\valuefootnoteapproxerrorbound">...
bounds<SUP>4.10</SUP></A>
<DD>
<PRE>.
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</PRE>
<DT><A NAME="fnm\valuefootnoteDEC5000">... example<SUP>4.11</SUP></A>
<DD>
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</PRE>
<DT><A NAME="foot13239">...
ill-conditioned</A><A NAME="foot13239"
 HREF="node90.html#tex2html2204"><SUP>4.3</SUP></A>
<DD>These bounds are special
cases of those in section&nbsp;<A HREF="node91.html#secnonsym">4.8</A>.
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</PRE>
<DT><A NAME="fnm\valuefootnoteapproxerrorbound">... bounds<SUP>4.10</SUP></A>
<DD>
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</PRE>
<DT><A NAME="fnm\valuefootnoteDEC5000">... example<SUP>4.11</SUP></A>
<DD>
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</PRE>
<DT><A NAME="foot13242">... eigenvalue.</A><A NAME="foot13242"
 HREF="node93.html#tex2html2253"><SUP>4.1</SUP></A>
<DD>Although such a one-to-one
correspondence between computed and true eigenvalues exists, it is not as
simple to describe as in the symmetric case. In the symmetric case the eigenvalues
are real and simply sorting provides the one-to-one correspondence,
so that 
<!-- MATH
 $\hat{\lambda}_1 \leq \cdots \leq \hat{\lambda}_n$
 -->
<IMG
 WIDTH="108" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img525.gif"
 ALT="$\hat{\lambda}_1 \leq \cdots \leq \hat{\lambda}_n$">
and 
<!-- MATH
 $\lambda_1 \leq \cdots \leq \lambda_n$
 -->
<IMG
 WIDTH="108" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img579.gif"
 ALT="$\lambda_1 \leq \cdots \leq \lambda_n$">.
With nonsymmetric matrices

<!-- MATH
 $\hat{\lambda}_i$
 -->
<IMG
 WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img530.gif"
 ALT="$\hat{\lambda}_i$">
is usually just the computed eigenvalue closest to
<IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.gif"
 ALT="$\lambda_i$">,
but in very ill-conditioned problems this is not always true.
In the most general case, the one-to-one correspondence may be described
in the following nonconstructive way: Let <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.gif"
 ALT="$\lambda_i$">
be the eigenvalues
of <B><I>A</I></B> and 
<!-- MATH
 $\hat{\lambda}_i$
 -->
<IMG
 WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img530.gif"
 ALT="$\hat{\lambda}_i$">
be the eigenvalues of <B><I>A</I>+<I>E</I></B>. Let
<IMG
 WIDTH="40" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img580.gif"
 ALT="$\lambda_i(t)$">
be the eigenvalues of <B><I>A</I>+<I>tE</I></B>, where <B><I>t</I></B> is a parameter,
which is initially zero, so that we may set 
<!-- MATH
 $\lambda_i (0) = \lambda_i$
 -->
<IMG
 WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img581.gif"
 ALT="$\lambda_i (0) = \lambda_i$">.
As <B><I>t</I></B> increase from 0 to 1, <IMG
 WIDTH="40" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img580.gif"
 ALT="$\lambda_i(t)$">
traces out a curve from
<IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.gif"
 ALT="$\lambda_i$">
to 
<!-- MATH
 $\hat{\lambda}_i$
 -->
<IMG
 WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img530.gif"
 ALT="$\hat{\lambda}_i$">,
providing the correspondence.
Care must be taken when the curves intersect, and the correspondence
may not be unique.
<PRE>.
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</PRE>
<DT><A NAME="fnm\valuefootnoteapproxerrorbound">...
bounds<SUP>4.10</SUP></A>
<DD>
<PRE>.
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</PRE>
<DT><A NAME="fnm\valuefootnoteDEC5000">... example<SUP>4.11</SUP></A>
<DD>
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</PRE>
<DT><A NAME="foot13254">...
<!-- MATH
 $\{\hat{v}_i \, , \, i \in {\cal I}\}$
 -->
<IMG
 WIDTH="87" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img666.gif"
 ALT="$\{\hat{v}_i \, , \, i \in {\cal I}\}$"></A><A NAME="foot13254"
 HREF="node97.html#tex2html2331"><SUP>4.1</SUP></A>
<DD>These bounds are special
cases of those in sections&nbsp;<A HREF="node89.html#secsym">4.7</A> and <A HREF="node91.html#secnonsym">4.8</A>,
since the singular values
and vectors of <B><I>A</I></B> are simply related to the eigenvalues and eigenvectors of
the Hermitian matrix 
<!-- MATH
 $\left( \begin{array}{cc} 0 & A^H \\A & 0 \end{array} \right)$
 -->
<IMG
 WIDTH="100" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img667.gif"
 ALT="$\left( \begin{array}{cc} 0 &amp; A^H \\ A &amp; 0 \end{array} \right) $">
[<A
 HREF="node151.html#GVL2">55</A>, p. 427].
<PRE>.
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</PRE>
<DT><A NAME="fnm\valuefootnoteapproxerrorbound">... bounds<SUP>4.10</SUP></A>
<DD>
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</PRE>
<DT><A NAME="fnm\valuefootnoteDEC5000">... example<SUP>4.11</SUP></A>
<DD>
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</PRE>
<DT><A NAME="foot13257">... magnitude</A><A NAME="foot13257"
 HREF="node99.html#tex2html2412"><SUP>4.1</SUP></A>
<DD>This bound is guaranteed
only if the Level 3 BLAS are implemented in a conventional way,
not in a fast way as described in section <A HREF="node108.html#secfastblas">4.13</A>.
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</PRE>
<DT><A NAME="foot13260">...
large</A><A NAME="foot13260"
 HREF="node99.html#tex2html2413"><SUP>4.2</SUP></A>
<DD>Another interpretation of chordal distance is as half the usual
Euclidean distance between the projections of 
<!-- MATH
 $\hat{\lambda}_i$
 -->
<IMG
 WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img530.gif"
 ALT="$\hat{\lambda}_i$">
and
<IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.gif"
 ALT="$\lambda_i$">
on the Riemann sphere, i.e., half the length of the chord
connecting the projections.
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</PRE>
<DT><A NAME="fnm\valuefootnoteapproxerrorbound">... bounds<SUP>4.10</SUP></A>
<DD>
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</PRE>
<DT><A NAME="foot12096">... suppose</A><A NAME="foot12096"
 HREF="node100.html#tex2html2431"><SUP>4.1</SUP></A>
<DD>This numerical example is computed
in IEEE single precision arithmetic on a SUN Sparcstation 10 workstation.
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</PRE>
<DT><A NAME="foot13385">...
pair.</A><A NAME="foot13385"
 HREF="node102.html#tex2html2442"><SUP>4.2</SUP></A>
<DD>As for the nonsymmetric eigenvalue problem there is an
one-to-one correspondence between computed and exact eigenvalue pairs,
but the correspondence is not as simple to describe as in the generalized
symmetric definite case. (Since the eigenvalues are real, sorting provides
the one-to-one correspondence). For well-conditioned generalized eigenvalues

<!-- MATH
 $({\hat{\alpha}}_i, {\hat{\beta}}_i)$
 -->
<IMG
 WIDTH="58" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img727.gif"
 ALT="$({\hat{\alpha}}_i, {\hat{\beta}}_i)$">
is usually the closest eigenvalue pair
to 
<!-- MATH
 $({\alpha}_i, {\beta}_i)$
 -->
<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img725.gif"
 ALT="$({\alpha}_i, {\beta}_i)$">
in the chordal metric,
but in ill-conditioned cases this is not always true. The (nonconstructive)
correspondence between computed and exact
eigenvalues described in footnote&nbsp;1 for the standard nonsymmetric eigenvalue problem is also applicable here.
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<DT><A NAME="foot13386">... eigenvalues</A><A NAME="foot13386"
 HREF="node102.html#tex2html2443"><SUP>4.3</SUP></A>
<DD>In order to make
the definition 
<!-- MATH
 $( {\alpha}_{\cal I}, {\beta}_{\cal I} )$
 -->
<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img763.gif"
 ALT="$( {\alpha}_{\cal I}, {\beta}_{\cal I} )$">
of the cluster average in chordal sense meaningful
for all cases that can appear we have to make a normalization of the
matrix pair in generalized Schur form.
First, we make each nonzero <B><I>b</I><SUB><I>ii</I></SUB></B> nonnegative real
by premultiplying by the appropriate complex number with unit absolute value.
Secondly, if all <B><I>b</I><SUB><I>ii</I></SUB></B> in the cluster are zero, then
we make all real parts of each nonzero <B><I>a</I><SUB><I>ii</I></SUB></B> nonnegative real.
This means that if there is at least one finite eigenvalue in the
cluster (i.e., at least one <B><I>b</I><SUB><I>ii</I></SUB></B> is nonzero in <B><I>B</I><SUB>11</SUB></B>), then

<!-- MATH
 ${\rm trace}(B_{11}) = \sum_{i \in {\cal I}} {\beta}_i$
 -->
<IMG
 WIDTH="164" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img764.gif"
 ALT="${\rm trace}(B_{11}) = \sum_{i \in {\cal I}} {\beta}_i$">
is nonzero and
the cluster average is finite.
If all <B><I>b</I><SUB><I>ii</I></SUB></B> are zero and some <B><I>a</I><SUB><I>ii</I></SUB></B> is nonzero then

<!-- MATH
 ${\rm trace}(A_{11}) = \sum_{i \in {\cal I}} {\alpha}_i$
 -->
<IMG
 WIDTH="165" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img765.gif"
 ALT="${\rm trace}(A_{11}) = \sum_{i \in {\cal I}} {\alpha}_i$">
is nonzero
and the cluster average is infinity.
Note the pencil is singular if one pair 
<!-- MATH
 $(a_{ii}, b_{ii}) = (0,0)$
 -->
<B>(<I>a</I><SUB><I>ii</I></SUB>, <I>b</I><SUB><I>ii</I></SUB>) = (0,0)</B>
or close to singular if both <B><I>a</I><SUB><I>ii</I></SUB></B> and <B><I>b</I><SUB><I>ii</I></SUB></B> are tiny
(see Section <A HREF="node105.html#sec_singular">4.11.1.4</A>).
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</PRE>
<DT><A NAME="fnm\valuefootnoteapproxerrorbound">...
bound<SUP>4.10</SUP></A>
<DD>
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<DT><A NAME="fnm\valuefootnoteDEC5000">... example<SUP>4.11</SUP></A>
<DD>
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</PRE>
<DT><A NAME="foot19489">... output)</A><A NAME="foot19489"
 HREF="node113.html#tex2html2527"><SUP>5.1</SUP></A>
<DD>
(Input or output) means that the argument may be either an input argument or
an output argument, depending on the values of other arguments;
for example, in the xyySVX driver routines, some arguments are used either
as output arguments to return details of a factorization, or as input
arguments to supply details of a previously computed factorization.
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</PRE>
<DT><A NAME="foot19490">... (workspace/output)</A><A NAME="foot19490"
 HREF="node113.html#tex2html2528"><SUP>5.2</SUP></A>
<DD>
(Workspace/output) means that the argument is used principally as a work
array, but may also return some useful information (in its first 
element)
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</PRE>
<DT><A NAME="foot20821">... DREAL.</A><A NAME="foot20821"
 HREF="node130.html#tex2html2580"><SUP>6.1</SUP></A>
<DD>Changing
DBLE to DREAL must be selective, because instances of DBLE with an <EM> integer</EM>
argument must <EM> not</EM> be changed. The compiler should flag any instances
of DBLE with a COMPLEX*16 argument if it does not accept them.
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</PRE>
<DT><A NAME="foot21108">...
matrix.</A><A NAME="foot21108"
 HREF="node137.html#tex2html2713"><SUP>7.1</SUP></A>
<DD>The requirement is stated ``LDA <IMG
 WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img913.gif"
 ALT="$\geq$">
max(1,N)'' 
rather than simply
``LDA <IMG
 WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img913.gif"
 ALT="$\geq$">
N'' because LDA must always be at least 1, even if N = 0, 
to satisfy the 
requirements of standard Fortran; on some systems,
a zero or negative value of LDA would cause a run-time fault.
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</PRE>
</DL><ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
</ADDRESS>
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